Global Positioning System Reference
In-Depth Information
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The first one is that we might have to test many sets b i if the variances of the real-
valued ambiguities are large. Let d j denote the range for ambiguity j based on the
estimated variance. If we form sets b i for all possible combinations, then there are
Π
d i
such sets, i
1 ,...,n . An efficient algorithm is needed to shorten the computation
time for ambiguity fixing. If ambiguities can be successfully fixed with just one epoch
of observation, then the distinction between static and kinematic relative positioning
becomes less relevant; the economic benefits of such a rapid survey technique are
obvious. In addition, cycle slips would be rendered harmless because new ambiguities
could be fixed every epoch. Much effort has gone into optimizing the computational
approaches. The LAMBDA method has emerged as the favored method.
The second problem is that several candidate sets might pass the test (7.139).
Naturally, one would like to identify the correct candidate as soon as possible and
in doing so minimize the observation time. Discernibility of the candidate sets will
be addressed in Section 7.8.3.
=
[27
7. 8.1 Early Efforts
Lin
1.2
——
No
PgE
Given the float solution and the respective covariance matrix, Frei and Beutler (1990)
suggest a specific ordering scheme for the candidate ambiguity sets. The efficiency
of their algorithms relies on the fact that if a certain ambiguity set is rejected, then a
whole group of sets is identifiable that will also be rejected and consequently need
not be computed explicitly. Euler and Landau (1992) and Blomenhofer et al. (1993)
point out that the matrix L 22 in (7.136) remains the same for all candidate sets. They
further recommend computing (7.138) in two steps. If
L 22 b
b
[27
g
=
(7.140)
v T Pv can be written as
then
n
v T Pv
g T g
g i
=
=
(7.141)
i
=
1
As soon as the first element g 1 has been computed, it can be squared and taken as the
first estimate of the quadratic form. Note that
g 1
is substituted in (7.139) to compute the test statistic, which is then compared with the
critical F value. If that test fails, the respective trial ambiguity set can immediately
be rejected. There is no need to compute the remaining g i values. If the test passes,
then the next value, g 2 , is computed and the test statistic is computed based on
v T Pv
g 1 . The value
v T Pv
=
g 2 . If this test fails, the ambiguity set is rejected; otherwise, g 3 is
computed, etc. This procedure continues until either the zero hypothesis is rejected
or all g i are computed and the complete sum of the g square terms is known. This
strategy can be combined with the ordering scheme mentioned above.
Chen and Lachapelle (1995) take advantage of the fact that integer ambiguity
resolution accelerates if the number of candidates d i for ambiguity i is small. The
smaller these search ranges, the fewer ambiguity sets need to be tested. Their method
v T Pv
g 1 +
=
 
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